Feigenbaum number

With this identification, the stable stationary-state behaviour (found for the cubic model with 1 < A < 4) corresponds to oscillations for which each amplitude is exactly the same as the previous one, i.e. to period-1 oscillatory behaviour. The first bifurcation (A = 4 above) would then give an oscillation with one large and one smaller peak, i.e. a period-2 waveform. The period doubling then continues in the same general way as described above. The B-Z reaction (chapter 14) shows a very convincing sequence, reproducing the Feigenbaum number within experimental error. 

The universal convergence factor 4.69920. .. is known as the Feigenbaum number in honor of its discoverer (Feigenbaum, 1979). It is a transcendental number like jr or e—the first to be found in modem times ... 

This route should already be familiar to us from our discussion of the logistic map in chapter 4, Prom that chapter, we recall that the Feigenbaum route calls for a sequence of period-doubling bifurcations pitchfork bifurcations versus the Hopf bifurcations of the Landau-Hopf route) such that if subharmonic bifurcations are observed at Reynolds numbers TZi and 7 2, another can be expected at TZ determined by... 

S. lb the astonishment of scientists, the value of S turned out to be universal , i.e. characteristie for many very d erent mathematieal problems and, therefore, reached a status similar to that of the numbers -it and e. The numbers -it and e satisfy the exact relation —1, but so far no similar relation was found for the Feigenbaum constant. There is an appnmmate relation (used by physicists in phase transition theory) which is satisfied it + tan e = 4.669201932 <= 6. 

Lederberg J, Sutherland GL, Buchanan BG, Feigenbaum EA, Robertson AV, Duffield AM, Djerassi C (1969 Application of Artificial Intelligence for Chemical Inference — I — The Number of Possiole Organic Compounds - Acyclic Structures Containing C, H, 0, and N. J Am Chem Soc 91 2973... 

A growing number of surveys of existing synthesis mechanisms and systems is being published let s just mention those of [Barr and Feigenbaum 82], [Biermann etal. 84b], [Partsch and Steinbruggen 83], [Smith 84], [Goldberg 86], [Feather 87], [Lowry and Duran 89], [Steier and Anderson 89], [Biermann 92], [Bundy 92], [Sammut 93], and [Deville and Lau 94]. 

One-dimensional maps with a single extremum are predicted to exhibit dynamical behavior that is universal, that is, independent of the details of the map [48-50]. (Proof of universality requires only that the map be basically well-behaved consult [49] for technical details.) The best known prediction is that the periodic state should., with a change in bifurcation parameter, should lose its stability to a state with twice the period, and the latter state in turn should lose stability to a state with "period 4," and so on. FEIGENBAUM [48-49] showed that the convergence rate for the period doubling sequence is given asymptotically by a universal number, 6=4.669 that is, the interval in bifurcation parameter over which a state with period would occur should be 6... 

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